The same $150 million portfolio, run through three accepted value at risk models on the same day, produced loss estimates of $1.6 million, $2.4 million, and $2.5 million. Nobody made an arithmetic error. The models simply disagreed about what tomorrow looks like.
Value at risk (VaR) is the minimum loss expected a stated percentage of the time over a stated horizon, given assumed market conditions. Three elements are always present: a currency or percentage amount, a probability threshold, and a time horizon. "The 5% one-day VaR is €2.2 million" means that on roughly 5% of trading days, about one day a month, losses would be at least €2.2 million.
TRAP: VaR is a minimum loss, not a maximum and not an expected loss. "There is a 5% chance of losing €2.2 million" is wrong. The most you can lose in an unlevered portfolio is all of it.
A 5% VaR equals a 95% confidence level. Under a normal distribution, the 5% cutoff sits 1.65 standard deviations below the expected value, the 1% cutoff sits 2.33 standard deviations...
Common mistakes
- Calling VaR the expected or maximum loss. In Example 1, $2,445,150 is the floor of the worst 5% of days, not the average outcome and not a cap.
- Annualizing the VaR number. Multiplying the $2,445,150 daily figure by √250 gives about $38 million and by 250 gives about $600 million. Both are wrong. Annualize the mean by 250 and the standard deviation by √250, then recompute: 0.096000 − 1.65 × 0.159883 = −0.167807, so the annual VaR is about $25.2 million.
- Mixing up the z-multiples. A 1% VaR uses 2.33 standard deviations, a 5% VaR uses 1.65, and a one standard deviation move is a 16% VaR. A 1% VaR does not mean a 1% price move.
Bottom line
- VaR definition: minimum loss expected a stated percentage of the time over a stated horizon, in currency or percent
- Z-multiples: 1.65 for 5% VaR, 2.33 for 1% VaR, 1.00 for 16% VaR (one standard deviation)
- Three methods: parametric assumes a distribution, historical replays actual returns, Monte Carlo simulates from a chosen distribution; all start with risk decomposition
- Options: parametric handles them poorly, historical and Monte Carlo handle them well
Exam shortcut
Read the stem for three tokens before touching a calculator: threshold, horizon, and portfolio value. If the vignette hands you asset weights, volatilities, and a correlation, it wants parametric VaR, so run mean, then volatility, then the z-multiple, then the dollar step, in that order.
The full lesson (about 3,206 words, 21 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- market risk
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